7.

Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another.

8.

Parallel planes are such as do not meet one another though produced.

9.

A solid angle is that which is made by the meeting, in one point, of more than two plane angles, which are not in the same plane.

10.

Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude.

11.

Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes.

12.

A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it, in which they meet.

13.

A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms.

14.

A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.

15.

The axis of a sphere is the fixed straight line about which the semicircle revolves.

16.

The centre of a sphere is the same with that of the semicircle.

17.

The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

18.

A cone is a solid figure described by the revolution of a rightangled triangle about one of the sides containing the right angle,

which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled; and if greater, an acute-angled cone.

19.

The axis of a cone is the fixed straight line about which the triangle revolves.

20.

The base of a cone is the circle described by that side containing the right angle, which revolves.

21.

A cylinder is a solid figure described by the revolution of a rightangled parallelogram about one of its sides, which remains fixed.

22.

The axis of a cylinder is the fixed straight line about which the parallelogram revolves.

23.

The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram.

24.

Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

25.

A cube is a solid figure contained by six equal squares.

26.

A tetrahedron is a solid figure contained by four equal and equilateral triangles.

27.

An octahedron is a solid figure contained by eight equal and equilateral triangles,

28.

A dodecahedron is a solid figure contained by twelve equal pentagons, which are equilateral and equiangular.

29.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles,

DEF. A,

A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel.

PROPOSITION 1.-Theorem.

One part of a straight line cannot be in a plane, and another part out of it.

If it be possible, let AB, part of the straight line ABC, be in a plane, and the part BC be out of it.

Construction. Because the straight line AB is in the plane, it can be produced in the plane (I. Post. 1). Let AB be produced to D; and let any plane be made to pass through the straight line AD, and turn about AD until it pass through the point C.

Demonstration. Because the points B and C are in this plane, the straight line BC is in it (I. Def. 7).

Therefore

1. The two straight lines ABC and ABD in the same plane, have a common segment AB (I. 11 Cor.),

which is impossible.

Therefore, one part of a straight line, &c. Q.E.D.

PROPOSITION 2.-Theorem.

Two straight lines which cut one another are in one plane; and three straight lines which meet one another in three points, are in one plane. Let two straight lines AB and CD cut one another in E; and let three straight lines AB, BC, and CD meet one another in three points B, C, and E.

B

Then AB and CD are in one plane, also the straight lines AB, BC, and CD are in one plane.

Construction. Let any plane pass through the straight line AB, and let the plane be turned about AB until it pass through the point C.

C

Demonstration. Because the points E and C are in this plane,

therefore

1. The straight line CD is in it (I. Def. 7, and XI. 1), but the straight line AB is in the same plane (constr.); therefore 2. The straight lines AB and CD are in one plane. Again, because the points B and C are in the same plane with the point E in AB; therefore

1. The straight line BC is in this plane (I. Def. 7). But it has been proved that the straight lines AB and CD are in it; therefore

2. The three straight lines AB, BC, and CD, are in one plane. Wherefore, two straight lines, &c. Q.E.D.

PROPOSITION 3.-Theorem.

If two planes cut one another, their common section is a straight line.

Let two planes AB and BC cut one another, and let the line BD be their common section.

Construction. If BD be not a straight line, from the point D to B, draw, in the plane AB, the straight line DEB (I. Post. 1), and in the plane BC, the straight line DFB.

Demonstration. Because the two straight lines DEB and DFB have the same extremities B and D, and do not coincide, therefore 1. DEB and DFB include a space between them,

which is impossible (I. Ax. 10). Therefore

2. BD, the common section of the planes AB and BC, must be in a straight line.

Wherefore, if two planes, &c. Q.E.D.

PROPOSITION 4.-Theorem.

If a straight line stand at right angles to each of two straight lines at the point of their intersection, it is at right angles to the plane in which they are.

Let the straight line EF stand at right angles to each of the straight lines AB and CD, at E, the point of their intersection.

Then EF is at right angles to the plane of AB and CD, that is, the plane in which they are,

Construction. Make EA, EB, EC, and ED all equal to one another (I. 3). Through E draw, in the plane of AB and CD, any straight line GH. Join AD and CB, and let GH meet them in G and H. From any point F, in EF, draw FA, FG, FD, FC, FH, and FB.

Demonstration. Because the two sides AE and ED are equal to the two sides BE and EC, each to each, and they contain equal angles AED and.BEC (I. 15), therefore

T